As an American, I will be writing this from the perspective of someone from the USA. However, you'll find that while some of the particulars in how to prepare for a trip at home are different in different countries, once you're abroad, the principles are all pretty much the same.
Here's what I recommended that you carry with you:
-Credit and Debit Cards with Good Foreign Exchange Rates (One Visa and One MasterCard at least). This is the modern way to pay for things internationally. Visa and MasterCard debit cards work in ATMs in major cities across almost every country in the world, including Syria, Ethiopia, and other places where, if you search online, you'll find warnings claiming they don't work. If you've read more of my blog, you know how many countries I've traveled across. The only one that wouldn't let me take money out of an ATM was Rwanda, and that was only because I had a MasterCard. Visa works in ATMs there just fine. This just meant I had to go inside a Rwandan bank for a withdrawal, plus a fee. When you use your card in a local ATM, you will always have the option to operate the ATM in English, and you will always be given local currency (unless you specifically ask for something else).
Here's the big catch: most major banks slap massive fees on these transactions. Chase, for example, charges $3 for each withdrawal, plus 5% commission on all withdrawals. You can do a lot better than that.
How? By researching better options. I recommended getting a credit card and small checking account with a smaller bank or a credit union, making sure to ask them about what they charge for foreign transactions. Since I'm from Seattle, I use Boeing Employees Credit Union. Last time I checked, their credit card charged 1% on foreign charges, and their debit card charged 1% on ATM withdrawals. Much better.
In fact, if you look hard enough, you might be able to do better than that. For example, if you or a close family have any ties with the military, you can apply for a checking account with USAA-- not only are there no fees for foreign withdrawals, but they refund any other fees charged by the bank you draw the money from. That's unheard of and amazing. Rumor has it Charles Schwab has a similar deals for people considering opening up new accounts with them (the hope being that you use their investment services when you get home, since you'll already have an account with them).
Once again, make sure you have at least one each of Visa and MasterCard, between your credit and debit cards. Visa seems to be slightly more common abroad, but really you should be fine with either. American express will work in a handful of other countries, not very reliably. Discover card is not going to help you.
Finally, don't keep both cards in the same place. If what you keep one in is stolen, you're probably going to need the other one right away.
-American Dollars. This applies to citizens of other countries as well. American Dollars are the closest thing there is to international currency. Euros are the perpetual second, and also not a bad thing to have around in cash. British Pounds are a distant third. I always kept a stash of US$200 for tight spots. Part of it went in my money belt, the other part in a hidden place in my backpack. You will end up spending this at times. The way you can replenish it depends on which countries you're going through. Sometimes it makes sense to change your leftover currency back into dollars, but more often than not it's better to spend it on something useful and withdraw local currency in the next country from an ATM, which will probably be located very near the border. In several countries you will find that you can withdraw American dollars from the ATMs (Hong Kong, Panama, El Salvador, Nicaragua, Cambodia, and Ecuador all come to mind).
-Phone Numbers to Contact Your Bank Internationally. Most banks offer numbers that will accept international collect calls for travelers in tight spot. You'll need this in emergencies. Keep it where you keep other documents, like your travelers insurance (you do have travelers insurance, don't you?)
And that's it!
...but wait, some of you are saying. What about travelers checks? The answer is, in my opinion, don't bother. They used to be the way to carry money around abroad. And they do still have more security than cash-- if they're stolen you can get a refund, provided you still have the stubs, and they can't be used by anyone but you. The problem is, nobody uses them anymore. This means that in 90% of the world they aren't accepted for anything, and in the remaining 10%, the exchange rates you get for them are terrible. Add to the fact that most places charge you a fee to even issue them, and I think you have a losing proposition overall. At best, they're a good emergency stash. I tried using them in India, and I don't plan on ever using them again.
Oh and one more item. Some people talk about pre-paid ATM Visa cards. They go by a variety of names. They sound like a neat idea. Basically it's a disposable ATM card with a set amount of cash tied to it. The problem is that the ones I've seen get even worse rates than you get from major banks for debit cards. Maybe there are better ones now, but I have yet to see them. Do a little research. If they charge you more than 3% on foreign transactions, or charge you any kinds of flat fees for anything, don't use them.
And that's a wrap! I may follow this up with more in depth tips for money management while abroad (how to most effectively have a parent or other highly trusted person to help at home, plus online banking). But this should help with the access-to-cash question.