What is a cheap but good cut of beef for grilling?

I've had great results with flank steak.

Best to marinate it with a mix of salt, sugar/honey, vinegar/lemon/lime, water/whiskey/tequila/soda, olive oil, spices, and herbs for at least a few hours before grilling. More salt will help tenderize but can't be left too long at too high concentrations (search "brining").

Don't overcook it, of course, that's the quickest way to make it chewy (unless you're going to cook it on low heat for several hours). Then let it rest before eating.

For flank steak, cut it across the grain. If you look closely the meat has fibers which run parallel to eachother in only one north-south direction. Cut across this, severing many lines with your slice. It severs the connective tissues before chewing, doing a lot of the work with the knife rather than your teeth.

I notice Flank steak already made the list, and once I would have agreed, but it's gotten trendy, and with the trendiness, expensive. It's still not fillet-price, but it tends to run in the 7-8 dollar a pound range where I live.

The steak I like to grill that I find to be tasty, cheap, and available is skirt steak. Do a light marinade, grill it as lightly as possible, cut across the grain of the meat, serve.

Your mileage may vary, however. The cheapest cuts are almost always the ones that local cooks don't know what to do with, and if the locals are hounds for the fajita, you're not going to be able to find skirt steak at all.

My advice is to go to the store with an open mind, and browse the meat counter looking for deals. I came home with 3 pounds of tenderloin tips for less than twenty bucks the other day: if I'd gone out looking to buy such a thing, I'd have been disappointed.

You should be able to get good results with any steak cut: ribeye, porterhouse, strip, sirloin, t-bone.

If you can't, there is something wrong with your process that needs to be fixed.

Make sure the steak is all the way thawed.
Either coat the steak with salt for 1 hour before grilling, or marinate in an acidic liquid (lemon juice, wine, marinade with vinegar).
Heat the grill very hot, then put the steaks on.
Use tongs, not fork or knife to manipulate steaks.
When proper sear lines have developed, rotate the steak 90 degrees.
When sear lines develop again, flip steaks over.
When sear lines develop on the second side, test for doneness by feel. If the steaks aren't done yet (for the thicker cuts), move the steaks to a cooler part of the grill and finish with indirect heat.
Put steaks on a plate on top of a hot pad and cover with a loose piece of foil and let them sit for 5 minutes. The internal temperature of the steak will go up a few degrees during this time.

(note -- US names for cuts of meat)

Part of the 'tenderness' of the meat comes in how you cut the meat. In "London Broil", you use an slightly tougher cut of meat (I normally use bottom round), grill it, then thinly slice it across the grain before serving.

You can do the same thing with flank steak, but I find it to be overpriced in the general grocery stores.

I like Rib Steaks.
They have a lot more fat, and you have to be careful about flare up, but they cook up really nice.