What price range do you consider cheap? If you're talking less than $500 out the door (without scope) then I have the following observations.
I have the following "cheap" new hunting rifles, Mossberg 4x4, Marlin XL7 & XS7, TC Venture, and the 700 ADL in various calibers. Out of those, IMO the TC Venture is the best of the bunch for quality, accuracy, fit/feel/finish, and price/value. Followed by the Marlins, Remington and last Mossberg.
I wouldn't hesitate to bring any of them on a hunting trip but after 50-100 rounds through each gun (to find best handload) at the range, the TC Venture in .270 is what I really like. If I knew what I know now, I wouldn't have bought the Remington & Mossberg and put that $ towards the new Vanguard S2 and a nice scope.
Good luck and have fun in your search.