One important point often not covered in textbooks and websites about Japanese are the notions of accent and stress. In general, Japanese language sounds flat as there are no stress accents like in English, but in fact there are tones to words, and if you don’t learn some basic tips, native speakers may get confused.
Here’s an example: the words ame (雨 rain) and ame (飴 candy) are pronounced the same, and written the same in hiragana. However, the tone is different. It took me a long, long time to realize this, and it’s almost never explained in Japanese courses (even for the JLPT). It’s something only a native speaker could explain.
To give you an idea of the difference, the word for rain sounds like AH-meh, while the word for candy sounds like ah-MEH. In the first example, the first syllable is more higher in tone. So, it starts on a higher tone, but goes down. In the second example, the second syllable is stressed and higher in tone. If you can, get a native speaker to point out the difference. My little girl has an Anpanman toy that teaches language, and one of the games tests listening on works like ame and ame, so I was embarrassed when I realized that I couldn’t tell the difference at first.1
Other examples include: KI-ru (切る to cut) and ki-RU (着る to wear); HA-shi (箸 chopsticks) and ha-SHI (橋 bridge).
But other words that are not homophones can have special stress on them. For example when asking how much something costs, or o-ikura (お幾ら), the “I” changes in tone, so it sounds like o-I-kura.
Because in American English we stress the second syllable, rather than focus on tones, our Japanese may sound funny.2 For the girl’s name Masako, in Japanese sounds like ma-sa-ko, while in American English I’ve heard it pronounced incorrectly as ma-SAH-ko. My wife’s name is similarly mispronounced. It’s not really wrong, but it’s just that the tone and stress is wrong.
So, as you study Japanese, try to watch for tones and stressing of syllables, and spend a little time reciting the words yourself to get it right. When you’re in a situation where you have to think fast, that little extra bit of practice can get you out of a difficult situation. Even if you studied a language for a long time, if someone can’t understand your pronunciation, how do you expect to communicate? 
Update: Robert at shiawase.uk has a nice post on tones in Japanese here.
Update #2: Made further corrections and clarifications utilizing advice from Robert. Thanks much!
1 Actually I struggle with her language toy in general. When you insert a new game, it asks you to push a certain button, and it took me days to realize what it was asking. Also, some of the basic vocabulary words and colors on there were words I didn’t know. My 2-year old has no trouble however. :p
2 One of my co-workers is Polish and he told me that in Polish the penultimate (second-to-last) syllable is always stressed in Polish, so if you keep that in mind you can speak Polish well. Other languages no doubt have their own rules.
This blog is the reflection of my efforts to explore fatherhood, Buddhism and my love of Japanese culture. Anything useful I can pass along makes the Internet hopefully a better place.
I would say that the *tone* changes, but there is no stress if you see the difference.
English uses stress accent, Japanese uses rising and falling tones.
The problem is, it isn’t consistent. there are regional changes and also changes depending on context, the words around a given word can change the tonal accent. I think this is one reason why L2 textbooks tend not to address it.
Interestingly it can also be a problem for Japanese. If you ever read The Japanese Language by Haruhiko Kindachi
I came across a set of general rules once I must see if I can find them again. –I’m not sure that the first syllable is always accented.
Like ロバート said, it’s important to distinguish stress and accent. In English, a syllable under stress usually carries a tone, but it’s also louder and longer, with the other syllables around it being reduced to schwa (this is why English speakers often make misspellings like “should of” for “should have”). In Japanese, the pitch accent generally just changes the pitch contour of a word, with the other vowels not being reduced to schwa. A lot of work was done in this area years ago by the linguist Mary Beckman (herself a native Japanese speaker). Given that I haven’t seen much about Japanese pitch accent recently, I’m guessing that absence means the linguists think they’ve got it figured out. Here’s an interesting link to follow up on.
In one of my Japanese study books (Complete Course of Japanese Conversation-Grammar by Oreste and Elisa Enko Vaccari) uses accent marks when they are spelled out in romaji on the letters that should be stressed. So, for your first example, it would be written as áme and amé in order to show the differences. Most of the study material I come across does not do this though unfortunately since, as you said, it can be handy to know. Though I also realize it’s a pain to add in those accented characters.
Hi everyone, I haven’t had time to write back a thoughtful reply, but thank you all for your comments. It was a big help.