I highly recommend Eleanor Jorden's and Mary Noda's "Japanese: The Spoken Language" series. It does an excellent job of explaining grammar using linguistics to help you really understand the implications of the language.
This is precisely the reason I
do not recommend this book.
In addition to it being a book for linguists, it also uses a non-Hepburn romanization system which, though it should surely please linguists, is completely useless to everyone else.
Why is the romanization important if we plan on not being reliant on roumaji? Most all dictionaries and software that we use expect us to use romanizatins like
fuji, not
huzi...
The book
is in fact, quite dated, and I think should be avoided, unless you like your language learning to be as stolid & rote as possible. There aren't any additional media for the text, for listening comprehension excercises, etcetera. There are a lot of reasons not to like this text.
I recommend Genki, you can order it online and it generally receives pretty good reviews, it's also a fun text, and has extensive notes and workbooks, which are generally worth it.
There are other books that are similar, like Japanese for Everyone, and the more dated, Situational Functional Japanese, which seems to have been supplanted by Genki these days as far as the most recommend coursework.
I've not seen the book myself, but as said above, Minna no nihongo get's thrown around a bit these days too.
We need to sticky a thread just full of book recommendations, these questions tend to pop up quite frequently.