If I create an IDE Raid array using a particular IDE Raid controller
(Promise Fasttrack276/376), can I use the disks and recover the data
using other manufacturer's Raid controllers?

Most of my experience has been with SCSI RAID systems and you CANNOT
read a RAID array from created on one controller with a controller
made by a different manufacturer.  Sometimes even different models
from the same manufacturer can give problems.

I concur with gregaw.

Once you create your RAID, I think you're committed to that controller.

-JtB

I agree. 

The data on the disks will only be able to be read with compatible
controllers (although some Software RAID drivers have emulation modes
specifically for recovering data from dead hardware RAID arrays).

For this exact reason, many people recommend Software RAID as a better
option for most applications.

Regards,
Sycophant