You could create interfaces for animal classes (class in the biological meaning), such as public interface Equidae for horses and public interface Avialae for birds (I'm no biologist, so the terms may be wrong).
Then you can still create a
public class Bird implements Avialae {
}
and
public class Horse implements Equidae {}
and also
public class Pegasus implements Avialae, Equidae {}
In order to reduce duplicate code, you could create an abstract class that contains most of the common code of the animals you want to implement.
public abstract class AbstractHorse implements Equidae {}
public class Horse extends AbstractHorse {}
public class Pegasus extends AbstractHorse implements Avialae {}
I'd like to add one more detail. As Brian remarks, this is something the OP already knew.
However, I want to emphasize, that I suggest to bypass the "multi-inheritance" problem with interfaces and that I don't recommend to use interfaces that represent already a concrete type (such as Bird) but more a behavior (others refer to duck-typing, which is good, too, but I mean just: the biological class of birds, Avialae). I also don't recommend to use interface names starting with a capital 'I', such as IBird, which just tells nothing about why you need an interface. That's the difference to the question: construct the inheritance hierarchy using interfaces, use abstract classes when useful, implement concrete classes where needed and use delegation if appropriate.