Yes, Mexican chorizo sure is greasy. I would have to second (or third) the comment that Mexican chorizo goes well with breakfast. Eggs, hashbrowns, scrambled eggs & chorizo in a tortilla with whatever toppings you like. Yum yum!
For Mexican chorizo your deli may make a good chorizo too. You're always going to have grease, with Mexican chorizo, but one of my deli's does have better meat...and seasoning than the two other brands that are pre-made in the store. You should give your grocery stores a check and see if they make it fresh too.
Spanish chorizo (as stated above) is indeed an entirely different thing. The Palacios chorizo is a hard, dry cured chorizo...this goes well in rice dishes. I've also had a nice slicing chorizo, Cantimpalo style chorizo. This was more like a traditional deli type meat, on the lines of salami (but different). It's nice to just eat alone, on a sandwich or with a charcuterie plate.
Anyway you slice it...or cook it, you're sure to enjoy!