The two extremes are the most-likely sources of death for creatures in space - weightlessness and the g-force of takeoff.
Weightlessness could be a critical issue for any creature which relies totally on gravity for swallowing - it's likely that some bird species would not be able to properly eat or drink in space.
In the long-term, it's likely that a few weeks of weightlessness won't kill many terrestrial creatures, even if it causes some difficulty.
Crushing damage or heart failure from G-force are much more likely causes of death.
Small creatures are probably going to be fine - many small creatures regularly submit themselves to far higher G-force than takeoff... just by hopping around. Froghopper bugs hit ~400G regularly.
Fleas hit 100G's.
I'm sure house cats break 3G easily, and that's space flight.
The square-cubes law helps explain this - proportionate strength makes is more likely for small things to hit higher G-force in their daily existence.
As a back-of-the-envelope calculation, since we know that humans can survive, we can assume (pretty safely) that any creature smaller than a human will probably survive the G-force of takeoff.
So what about the big boys?
Elephants are unlikely to survive. An elephant at 3G is going to be experiencing ~36,000 pounds of force, although most elephants seem capable of handling roughly ~1.5G for a limited period of time - ask National Geographic about how that works.
36,000 pounds still sounds lethal to me.
Giraffes would die - their hearts are already taxed by 1G.
Whales are absolutely toast - they can't survive 1G out of water for long, you can forget about 3G in any circumstance.
I have no idea where the line will be drawn, and I imagine that other factors matter as well... but I think that anything larger than a grizzly bear is likely to suffer fatal complications from extended 3G forces. It's just more than their circulatory system can handle.
Some creatures may be able to survive an otherwise fatal journey if placed into a gel or other liquid substance, although the container would have to be immensely strong or perhaps also serve to immobilize them (good luck building a tank that can survive an angry whale.) I don't think this would prevent all circulatory issues, although it would serve to spread out pressure and reduce damage from crushing.
Ultimately, we're unlikely to know unless we test, and I'm not enthusiastic about the ethics of the tests that would be required.