Could you screw 1/2" drywall to the subfloor "between" the beams? Then either wrap each beam with drywall, or wrap them with lumber and stain (assuming the beams aren't attractive enough to leave exposed). Accent the beams instead of hiding them. Just watch the length of the screws so you don't have screws coming up through the floor. :) Of course, this approach would make it difficult to run wiring, piping, insulation, etc.
Alternatively, how far do the beams span? Maybe you could add a joist or two "parallel" to the beams instead of short segments between beams. The beams support the floor, so the new joists only need to be sized like ceiling joists to hold up the drywall. So, depending on the span, you could probably use a single 2x4 or 2x6 between the beams (I'd probably opt for two joists for a 16" OC spacing).
If you must run 2x4 blocking between the beams, mark out 16" spacing along the two end beams, then snap chalk lines across the bottoms of the beams so you'll know where the blocking needs to go.
Then take a few measurements at various places to find out what the largest distance is. Then use a power miter saw and a stop block to cut several blocks an inch or so longer than the longest you'll need.
Now grab a handful of the blocks, and go to one of the bays between the beams. Hold one end of the block against the side of one beam, and mark the exact length you'll need to cut the block. Write a number 1 on this block, and a number one on the side of the beam so you know where this block belongs.
Move to the next chalkline, and repeat the procedure, this time writing 2 on the block and beam. Repeat for the rest of the bay.
Take the blocks back to the miter saw and cut them all to the lengths you marked.
Now you should have blocks ready to nail up in that bay. They should be a snug fit, so tap them in with a hammer, centering over the chalkline. You can toenail them in by hand if you don't have an air nailer, or use a screwgun and 3" deck screws.
By marking the blocks in position, you'll avoid measuring errors and can accomodate fluctuations in the spacing between the beams.
You can cut the blocks with a circular saw or even a handsaw if you don't have a power miter saw. But this sounds like a good excuse to buy a new toy. :) Worst case, you could probably rent a miter saw from a tool rental store.
Have fun!
Anthony