please see if there is anyone using the name Flying Crane Productions 
- I want to start a company with that name.

Greetings, quibbory!

The Flying Crane is a popular motif in Japanese culture:
http://www.chopa.com/ShopSite/product157.html


There is a shop that does business as The Flying Crane in Frisco,
Colorado; they specialize in Japanese items "sake & tea, sushi & soup,
ikebana, and wall art".
http://www.theflyingcrane.com


There is also a circus group named The Flying Cranes which has
registered a trademark in the U.S.

"Word Mark THE FLYING CRANES 
Goods and Services IC 041. US 100 101 107. G & S: Entertainment
services in the nature of live flying trapeze performances.
FIRST USE: 19850300.
FIRST USE IN COMMERCE: 19881000 
Mark Drawing Code (3) DESIGN PLUS WORDS, LETTERS, AND/OR NUMBERS"
http://tess2.uspto.gov/bin/showfield?f=doc&state=4j8vl1.2.1

This in and by itself might not prohibit you from using "Flying Crane
Productions", providing that the nature of your business is
sufficiently different from that of The Flying Cranes.

I found this record using TESS (the US Patent and Trademark Office's
Trademark and Electronic Search System):
http://tess2.uspto.gov/bin/gate.exe?f=searchss&state=auaub9.1.1

I have not been able to find any reference at the USPTO, or anywhere
else online to a business named "Flying Crane Productions".


HOWEVER:
Simply not finding the name in a search on the site of the USPTO is
NOT considered legally sufficient protection in cases of
copyright/trademark infringement. To be totally protected, you will
need to have an experienced patent and trademark attorney do a search
for you. Such an attorney will know all the places that need to be
checked to provide an accurate verdict as to whether you can use that
name without infringing on someone else's trademark/copyright.

If such an attorney would be prohibitively expensive for you, you can
trying asking for help from the US Patent and Trademark Office. Their
website is here:
http://www.uspto.gov/main/trademarks.htm


Search Strategy

"flying crane" -sikorsky -hiller -percival -hughes -aircraft
://www.google.com/search?hl=en&lr=&ie=UTF-8&oe=UTF-8&q=%22flying+crane%22+-sikorsky+-hiller+-percival+-hughes+-aircraft&btnG=Google+Search


Before Rating this Answer, if you have any questions about the
information I have presented, please post a Request for Clarification,
and I will see what I can do for you.
 
I hope that this Answer provides you with the information you are
seeking. Best wishes on your new business!
 
Regards, 
 
aceresearcher

---

Clarification of Answer by aceresearcher-ga on 11 Mar 2003 11:43 PST

Well, geez.

The USPTO site apparently doesn't allow you to make permanent search
links. To duplicate my results, you will need to go to their site at
http://www.uspto.gov/main/trademarks.htm 

and click on
"* SEARCH trademarks (TESS)
(pending and registered trademarks)"

On the resulting page, click on
"New User Form Search (Basic)"

On the resulting search page, enter "Flying Crane" (withOUT the
quotes) in the box marked "Search Term:" and make sure that the
"Result Must Contain:" box specifies "All Search Terms (AND)".

If you have problems accessing the results this way, please post a
Request for Clarification, and I will try to assist you.

Regards,

aceresearcher

There's none registered in the UK ...

04645006    THE FLYING CHEF LIMITED      
04649494    FLYING CHEF EVENTS LIMITED      
04653486    FLYING CHILLI LIMITED      
03826073    FLYING CIRCUS LIMITED      
03277650    THE FLYING CIRCUS CLUB LIMITED      
03994304    THE FLYING CLASSROOMS LIMITED      
04163246  D THE FLYING CLEANERS LIMITED    Dissolved  
04387716    FLYING CLOUD LTD.      
SC233777    THE FLYING CLUB LIMITED      
04275388    THE FLYING COFFEE BEAN (UK) LTD      
SC194797    FLYING COLORS DECORATORS LIMITED      
SC238958    FLYING COLORS (SCOTLAND) LTD.      
03073519    FLYING COLOURS LIMITED      
03800271    FLYING COLOURS AIRLINES LIMITED      
03141402    FLYING COLOURS AVIATION LIMITED      
04578951    FLYING COLOURS BROCHURES LIMITED      
04583191    FLYING COLOURS COLOUR PRINTERS LIMITED      
03467614    FLYING COLOURS COMMUNICATIONS LIMITED      
04400398    FLYING COLOURS CONSTRUCTION LIMITED      
03342048    FLYING COLOURS CONTRACTS LIMITED      
03308776    FLYING COLOURS DESIGN LIMITED      
03456104    FLYING COLOURS DESIGN CONSULTANTS LIMITED      
SC151128    FLYING COLOURS (ENTERPRISES) LIMITED      
02259565    THE FLYING COLOURS FILM CO. LIMITED      
04672024    FLYING COLOURS FLAGMAKERS LIMITED      
03093628    FLYING COLOURS GROUP LIMITED      
03863793    FLYING COLOURS HOLDINGS LIMITED      
02559272    FLYING COLOURS HOLIDAYS LIMITED      
03124442    FLYING COLOURS LEISURE GROUP LIMITED      
04226828    FLYING COLOURS MOTORCYCLE TRAINING LIMITED      
04496513    FLYING COLOURS MOTORCYCLE TRAINING (NORTH) LIMITED      
02798869    FLYING COLOURS PHOTOGRAPHY LIMITED      
04177752    FLYING COLOURS PRAGUE LIMITED      
03163981    FLYING COLOURS RECRUITMENT SERVICES LIMITED      
03947905    FLYING COLOURS SCHOOL OF MOTORING LIMITED      
03656322    FLYING COLOURS (SERVICES FOR YOUNG PEOPLE) LTD      
03168875    FLYING COLOURS (SMART REPAIRS) LIMITED      
03330316  D FLYING COLOURS (SUSSEX) LIMITED    Dissolved  
03235625    FLYING COLOURS TRUSTEES LIMITED      
03985070    FLYING CROCODILE LIMITED      
04060041    THE FLYING CURRY LIMITED      
02966132    FLYING DISK-DOCTOR LIMITED      
03571592    FLYING DOCTOR COMPUTERS LIMITED      
00672253    FLYING DOCTOR DEVELOPMENT SERVICE LIMITED      
04201526    THE FLYING DOCTOR'S HOT AIR BALLOON COMPANY LIMITED      
03693992    FLYING DODO LIMITED      
03596252    FLYING DOG LIMITED      
04488125    FLYING DOG O P LIMITED      
04083107    FLYING DOG PRODUCTIONS LIMITED      
LP003547    THE FLYING DOLPHIN COMPANY      
04387677    FLYINGDOVE TECHNOLOGIES LIMITED      
SC208827    FLYING DRAGON LIMITED      
04447034    FLYING DRAGON PUBLISHING LIMITED      
03969991    FLYINGDRUM LIMITED      
03138915    FLYING DUCK LTD.      
04645362    FLYING DUCK MUSIC LIMITED      
04515712    THE FLYING DUCK PUB COMPANY LIMITED      
04066755    FLYING DUDES LTD      
02686885  D THE FLYING DUTCHMAN CO LTD.    Dissolved  
SC242668    FLYING DUTCHMAN ANTIQUES LTD.      
04560367    THE FLYING DUTCHMAN (QUEENBOROUGH) LIMITED      
03302977  D FLYING EAGLE ENTERPRISES LIMITED    Dissolved  
04100302    FLYINGEAGLENET LIMITED      
03770364    FLYING EAGLES LTD      
04232836    FLYING EAGLE SECURITY LIMITED      
04382601    FLYING EAGLES SERVICES LIMITED      
03487294    FLYING ELEPHANT FILMS LIMITED      
03980705    FLYING EYE LTD      
00590173    FLYING FACILITIES LIMITED      
03248251    FLYING FALCON AIR SERVICES LTD      
01166733    FLYING FARMERS ASSOCIATION LIMITED      
01404195  L FLYING FASHIONS LIMITED      
01182988    FLYING FINNS LIMITED      
03007425    FLYING FIRKIN DISTRIBUTION LIMITED      
03729406    FLYING FISH LIMITED      
04589502    THE FLYING FISH (BRENTWOOD) LIMITED      
03365666    FLYINGFISH DESIGN CONSULTANTS LIMITED      
03716857    FLYINGFISH FABRICS LIMITED      
04619059    FLYING FISH FEASTS LIMITED      
03480881    FLYING FISH FILM PRODUCTIONS LIMITED

The only US trademark I could find is Flying Cranes, Inc. An acrobatics troupe.
http://tess2.uspto.gov/bin/showfield?f=doc&state=eb3ae2.3.2

You can search the trademark office for US names here:
http://tess2.uspto.gov/bin/gate.exe?f=tess&state=eb3ae2.1.1