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.