They lived principally upon fish, venison and honey.
That's where our venison is coming from?
He dined on venison and champagne whenever he had been so fortunate as to borrow a guinea.
A forester was "an officer sworn to preserve the vert and venison in the forest, and to attend upon the wild beasts within his bailiwick."
To stay on the healthier end of the spectrum in regards to fat, try chicken and turkey, salmon and other (healthy) fatty fish, or even game meats like venison, which are lower in fat than beef.