It is "jíst rád", not "jít rád". It is common, but rád is an adverb there. Without "rád" it is also possible, but it means that the dogs eats some bones at all, not that he necessarily likes them too much.
It is also very common to use "mít rád" about food. More common than "jíst rád".
There is, žrát. But it seems people actually use jíst quite often anyway. At least for a dog. The Czech national korpus has 35 hits for "pes snědl" (the past tense of sníst, the perfective alternative of jíst) and 50 hits for "pes sežral" (the past tense for sežrat, the perfective alternative for žrát).