From what I have read , I would have to say no, religion did not have any influence in Greek and Roman morality, but it did have some influence in law. The Greeks and Romans understood their gods in a way that is very different to later Western civilization, and in some ways difficult for us to understand.

The Greeks and Romans genuinely feared their gods, and kind of saw them as big bullies. They made sacrifices literally in order to appease the gods so they wouldn't send a drought to destroy the harvest or a storm to wreck their ships. A good example of the human relationship to the gods can be seen in Ovid's epic poem, Metamorphoses (it can be found in translation via google). It's basically a collection of mythical stories about transformations. Many of them feature the gods doing terrible things to innocent people for basically no reason. Zeus goes around raping countless innocent girls, while his wife Juno punishes them out of jealousy (poor Io!). I don't know about you, but it doesn't seem like that behavior was intended to be a moral example.

Concerning the religion translated into law, I'm not as well versed but I can tell you what I know. Basically, the Roman state recognized the Greco-Roman mythology as the official "religion", but religion was not defined in the same way today. Important politicians were often appointed to ceremonially significant priesthoods, such as Julius Caesar becoming a member of the Flamen Dialis.