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I find that Japanese pitch accent seems to have some sort of interaction with the morphology. For example, 早い is pronounced [はやい]{LHL}, but 早く is pronounced [はやく]{HLL}.
Are these changes regular? No grammar book I have describes the interaction between accent and conjugation. I also find that the above pattern cannot be naively generalized ([すごい]{LHL} -> [すごく]{LHL}), but perhaps there is a deeper reason why the accent changes on はやい?
The changes are basically regular based on the "original accent" of each word, but (1) these "original accents" are not set in stone; (2) people/groups speak differently; and (3) pitch accent, like any linguistic phenomenon, is constantly changing. (I'm going to skip the discussion of whether accent exists, etc., and just stipulate that the last "high" mora before the drop is the "accent" of a given word.)
Most adjectives are like /hayai/ in that their "original accent" is two morae from the end. [よい]{HL}, [あおい]{LHL}, [うれしい]{LHHL}. In the /-ku/ form of these adjectives, the accent moves one morae back (or stays where it is if it's already on the first mora): [よく]{HL}, [あおく]{HLL}, [うれしく]{LHLL}.
But there are some adjectives with no "original accent" at all: [あかい]{LHH}, [かなしい]{LHHH}. These remain accentless in /-ku/ form: [あかく]{LHH}, [かなしく]{LHHH}.
Now the above is complicated by individual differences (perhaps influences from one's native dialect, etc.), group differences (old/young, Yamanote/Shitamachi, etc.), word differences (slang/formal, etc.) and just plain old language change.
For example, it seems to be becoming more common for adjectives with an "original accent" to keep the accent in the same place even when they are conjugated. Regarding すごい, for example, the Shinmeikai accent dictionary specifies [すごい]{LHL} and accordingly [すごくて]{HLLL}, but they have a note: "新は [すごくて]{LHLL}". So they recognize [すごくて]{LHLL} as a "new" form -- not "wrong" or even "informal", but "new".
No grammar book I have describes the interaction between accent and conjugation
Very few do! I am not sure why this is. It may be because very few books in Japanese about Japanese do this either, or at best give a cursory treatment at the back. This in turn may be because accent varies so wildly between dialects of Japanese, even where vocabulary, syntax, other-parts-of-phonology, etc. remain (relatively) stable.
I don't know of any really good way to learn Japanese pitch accent from English written materials. There are plenty of academic papers on it, of course, but they are more useful for reference than (self-)education -- learning about rather than learning. IIRC Samuel Martin's grammar marks accent scrupulously, but I don't remember how much explanation he offers and the book is prohibitively priced anyway.
However, in Japanese the situation is different. There are lots of books about accent available -- a nice recent one is Nihongo Akusento Nyumon (日本語アクセント入門), eds. Matsumori Akiko, Nitta Tetsuo, Kibe Nobuko, Nakai Yukihiko (Sanseido 2012). Some dictionaries also have accent information. The 日本国語大辞典 (available by subscription through JapanKnowledge) has information for multiple accents (Tokyo, Kyoto, historical). NHK publishes an accent dictionary (which its on-air personalities are expected to adhere to), and so does Shinmeikai. Both of those include information at the back explaining how accent is affected by conjugation and other phenomena.
ꜜ. Another user edited it into the red line format. I decided to edit the question and answer--hopefully that's okay with everyone :-)