I don’t think that’s such a great idea – the less romaji you see, the better. By learning kana, you’re trying to move your brain away from thinking in romaji (among other things). Best thing to do would be to get those all-kana textbooks you’re talking about (such as Japanese for Busy People I) and just try to read the words as best you can. If you come to a kana you are stuck with, whip out a handy, dandy little kana table and look it up; I’ve heard the process of looking something up rather than just seeing the answer helps you remember, which might have actually been said by Koichi in Textfugu now that I think about it.

Once you think you’re comfortable with your kana (both hira- and kata-), move onto starting on tackling kanji as soon as you can, it really does help.

“Bobby Boucher, that romaji’s the devil!”

I had a feeling you wouldn’t like that answer haha. In that case, the textbook Genki I has kana with romaji next to it; I’ve not read the whole thing, but it’s definitely like that at the start of the book. After a while, try finding something that’s purely kana – or native material with kana AND kanji, obviously focussing on the kana – and use the “look-up ones you don’t know” method. If the romaji is right there, your eyes will tend to look at it first even if you don’t mean to, which can lead you to think you’re doing better than you are, so make sure you practice without it too.

Don’t worry about “speeding up the process” though – most people I’ve seen online take a week or two (I took slightly longer) to become reasonably comfortable with both hiragana and katakana, so it really doesn’t take that long in the first place. And don’t underestimate the power of flash cards – they can be boring at times, but they’re really pretty effective. Who knows, you might even learn them faster that way :D