I've read both here. some say don't use anything but Old English to clean the grime of my 1898 stock, and some say BLO is what to use.

What do most of you guys use to clean these things?

The same principles used for cleaning/polishing any piece of finished wood apply (check around on the internet) with a few more likely. You'll need to determine what your finish is or, at the very least, how your finishes react to various cleaners/chemicals. Always test a
small out of the way area first. Some finishes will not melt/dissolve under most conditions - others easily start doing so (like shellac) and can we reworked nicely if you are prepared to do so. My bench usually includes various oils (do not discount WD-40), lacquer thinner, denatured alcohol, mineral spirits, and of course Old English. Different parts of the stock may need verious treatments especially around the action. As a final step, a bees wax polish may be in order - I use Birchwood Casey Stock Polish myself. Hopefully you don't have one finished with epoxy like I did - had to start over with that one.

I've had some success with a 50-50 mixture of linseed oil and turpentine (enough to about half fill a mayo jar) with a teaspoon of pumice mixed in. Apply with a terry-cloth rag.

I'll have to wait to do this until the weather cools off. I recently had heart surgery and have to recover another month or so.

50/50 BLO was the stock treatment of choice for many years.  The trouble with BLO it seems is that over time the stock turns Black.  I no longer use it on my pretty wood.  The stocks that are already black, well why not, it looks good  ;)