How do I pound chicken (or other meat) without making a mess?
Clingwrap works fine for me -- but I don't use a meat tenderizer -- I just use a small but fairly heavy pan (but not my cast iron, as it's not smooth on the bottom).

When I was in college, I tried a few things.  I can get pretty decent results just hitting it with my cutting board.  (with it between saran wrap).

Part of it might be technique -- if I'm doing chicken breast I first slice is sort of following the taper of the breast, so it's neither with or across the grain, but kinda of diagonal.  You also don't want to hit straight down -- you want to angle the blows, so you're actually pushing the meat out, and not just trying to force it through the counter.

And I should mention that I've only ever pounded out poultry and pork -- I assume beef would be the same, but I have no experience to confirm or deny it.