There are lots of Spanish furniture stores selling less "modern" goods. El Corte Inglés is a good place to start looking online, they have stores all over the country and will deliver.
Whether you easily will find what you are looking for depends on your tastes. The colonial/rustic style is very popular; you will have no problem if you want a Mexican hacienda look, or a Moroccan tea-room for that matter.
There are also many places that will make furniture to order, according to your taste. Labour costs are relative cheap here.
There seem to be less big chains like we have in the UK like DFS, Furnitureland, etc. but most reasonably sized retail parks will still have large furniture store in them. Also worth keeping an eye out for second hand furniture shops, they are quite popular especially in the more touristy areas, where landlords regularly use them to kit out cheap apartments. Theres a big one near me on the way to Marbella, not sure where in Spain you are though as your profile say 'Michigan'.
Oops. Thanks for reminding me I needed to update my location. I'm in Barcelona.
Venta Unica has a lot of stuff, and there are a few things I like there, thanks!
I guess I was just spoiled in the US. It was so easy finding "old world" or "European" inspired furniture there, for some strange reason. It really blows my mind how difficult it has been for me to find furniture I like here.
In case anyone is looking at this thread in the future... Maisons du Monde is probably the best / biggest place I have found for non-modern furniture. Of course, few of the online items are actually in any of their stores.