I have had this machine for 2 years, and lately it has started
intermittently failing.  It will play through a disk, or several
disks, and then will stop.  When I try to reload new disks, it fails
to recognize that a disk is there.  I have been able to get it to work
in the past by opening the case, jiggling a few wires, and putting it
back together again, but that does not seem to be working this time
(it crapped out on me last night halfway through Lord of the Rings). 
Please let me know if you know how to fix this.

bocksolibra,

Thank you very much for inviting me to post my Comments as an Answer!
I wish that I had been able to find a quick-and-easy repair solution
for you instead. My only hope is that, in the future, you can use the
resources I cited to avoid being victimized by crappy products.

It looks like many people have experienced similar problems with this
player:

http://www.amazon.com/exec/obidos/tg/stores/detail/-/electronics/B00004TZHN/customer-reviews/ref=cm_cr_dp_2_1/102-4743527-8712128.

http://www.audioreview.com/DVD+Players/Toshiba+SD-3205/PRD_124608_1587crx.aspx

You have my condolences :{ ... 
 
It sounds like a wire may be loose. If you know what you are doing,
you might be able to find the wire (most likely it has broken off
where it is soldered to the circuit board), re-strip the end, and
re-solder to the circuit board. Of course, this could also be a really
good way to completely fry your DVD player beyond repair (I am really
good at this sort of thing, the frying, I mean).
 
Unfortunately, because your player is out of warranty, it would
probably cost more to fix than it is worth, since DVD player prices
have gone down considerably in the last 2 years. If you decide to
replace it, try feeding in your desired characteristics ("DVD player
CD HDCD MP3" for example) into Amazon's Electronics Store Search
engine at

http://www.amazon.com/exec/obidos/tg/browse/-/172282/ref%3Dtab%5Fgw%5Fe%5F4/102-9843814-6836155
 
We just recently bought a combination DVD/VHS player. We used
http://www.circuitcity.com, http://www.bestbuy.com, and
http://www.amazon.com to search for models with the features we
desired, then checked the reviews for those items on all three
websites to decide which one we should purchase. (Another site to try
is http://www.epinions.com .) How fortunate that we did: the first
player we picked out had a pile of reviews similar to your Toshiba!
 
We walked into Circuit City, took a look at the model we had chosen,
paid for it and walked out within 5 minutes. I highly recommend this
method of research for any future purchase, especially electronics. It
(hopefully!) saved us from making a really bad mistake, and made the
whole purchase process hassle-free. Circuit City also provides the
ability to order and pay for your item online, then choose to have it
shipped to you OR go down to your local Circuit City the same day to
pick it up (provided it's in stock there)! No pesky salespeople!!!
 
Best wishes!

aceresearcher

It looks like many people have experienced similar problems with this
player:
http://www.amazon.com/exec/obidos/tg/stores/detail/-/electronics/B00004TZHN/customer-reviews/ref=cm_cr_dp_2_1/102-4743527-8712128.

It sounds like a wire may be loose. If you know what you are doing,
you might be able to find the wire (most likely it has broken off
where it is soldered to the circuit board), re-strip the end, and
re-solder to the circuit board. Of course, this could also be a really
good way to completely fry your DVD player beyond repair (I am really
good at this sort of thing, the frying, I mean).

Unfortunately, because your player is out of warranty, it would
probably cost more to fix than it is worth, since DVD player prices
have gone down considerably in the last 2 years. If you decide to
replace it, try feeding in your desired characteristics ("DVD player
CD HDCD MP3" for example) into Amazon's Electronics Store Search
engine at http://www.amazon.com/exec/obidos/tg/browse/-/172282/ref%3Dtab%5Fgw%5Fe%5F4/102-9843814-6836155.

Best wishes!

More bad news...

http://www.audioreview.com/DVD+Players/Toshiba+SD-3205/PRD_124608_1587crx.aspx

You have my condolences :{ ...

We just recently bought a combination DVD/VHS player. We used
http://www.circuitcity.com, http://www.bestbuy.com, and
http://www.amazon.com to search for models with the features we
desired, then checked the reviews for those items on all three
websites to decide which one we should purchase. How fortunate that we
did: the first player we picked out had a pile of reviews similar to
your Toshiba!

We walked into Circuit City, took a look at the model we had chosen,
paid for it and walked out within 5 minutes. I highly recommend this
method of research for any future purchase, especially electronics. It
(hopefully!) saved us from making a really bad mistake, and made the
whole purchase process hassle-free. Circuit City also provides the
ability to order and pay for your item online, then choose to have it
shipped to you OR go down to your local Circuit City the same day to
pick it up (provided it's in stock there)! No pesky salespeople!!!

Best wishes.

Another possibility...  

The laser optics used in modern DVD/CD players are mostly disposable
(read: dirt cheap).  Many tend not to last.  Typical the "sled" is a
closed package that contains loading mechanism and optics --- slap a
D/A converter and an output devices in and voila, a DVD player.  The
point is that if the laser is done, it is cheaper to buy a new player.

You can often buy extra life from a laser simply by cleaning the
optics (if accessible).  This is really simple --- take a cotton swap
and a methyl-alcohol based cleaning solution (available electronic
parts stores) and literally clean it like a window (gently though). 
We did this often at the repair shop I worked at in college... in 98
were charging about $25-$45 for the work.  Typically, it buys anywhere
from 3 mos to several years of life on the laser...  expect $50-$75
from a reasonable shop...

Good luck.

Sorry for the delay.  I had tried the lens cleaning trick a few times
in the past, and bought a couple more months, but now I think it's
good and dead.  I have written several times to Toshiba, but they
don't seem to care.  STAY AWAY from Toshiba products.

aceresearcher, go ahead and submit your stuff as the answer and claim
the cash if you want.

Thanks