In addition to what Jason says above, which is absolutely correct, nori needs to be toasted to be crispy.
Sometimes you buy it ready-toasted, but sometimes it's not toasted and you need to toast it over an open flame yourself.
How to toast nori instructions and video .
Additionally, I've found that if you live somewhere high-humidity (like San Francisco, where I live), you sometimes need to re-toast nori even if you're buying ready-toasted packets, because it becomes moist.
cooking.stackexchange.com/users/7180/FuzzyChef
Don't wait too long to eat it.
I'm assuming you're talking about rolls, or makimono, rather than nigiri-zushi.
Anyway, assuming the nori was reasonably freshly opened and crispy when you started, that's the main risk you have: within minutes of preparing makimono, the moisture of the rice will quickly transfer to the dry laver.
The faster you eat it, the less likely it is to become mushy and chewy.
In the convenience stores in Japan, onigiri, which are, like makimono, also wrapped in nori, are now usually wrapped in a special plastic pouch.
If you follow the directions, you can wrap the onigiri with the nori in one motion while you peel off the plastic.
This lets you enjoy the crispy texture when you're ready to eat, which is the one advantage convenience store onigiri have over bringing mom's with you.
If you make the onigiri at home, you'll usually end up with moist, slightly chewy nori when you're finally ready to eat.
cooking.stackexchange.com/users/2909/JasonTrue
As everyone has said, assuming the nori is dry enough to begin with, the problem is moisture getting into the nori from the rice.
But the problem might not be just moisture transferring because you let it sit too long, but rather that the rice itself is too moist.
The drying step in making sushi rice is critical, and if you under-dry your rice, you'll have problems.
Especially if your rice is pretty sticky, you may want to try drying it further.
From a great Serious Eats article about making the rice: If all goes well, your sumeshi should come out slightly sweet and tart, full of distinct grains, and have a texture that holds together when compressed, but is not overly sticky.
cooking.stackexchange.com/users/1672/Cascabel
To further the answers already given: Nori starts becoming chewy as soon as rice is applied to it since the rice has moisture thus you want to minimize the time between rice-on-nori to maki-in-mouth.
Besides toasting the nori and consuming the maki roll as soon as possible after producing it, I would recommend one step between those two: Mise-en-place; not only should you have all your materials/tools/ingredients at the ready, you should have each makimono filling pre-assembled BEFORE applying the rice.
For example, if you have a roll with multiple ingredients in the filling gather enough fillings for a single roll in one pile so it can all be added to the roll all at once.
Also make sure that you have a clean and dry area for assembling, cutting, and plating the maki roll.
Again, keep moisture away from the nori as much as possible.
cooking.stackexchange.com/users/15038/ceelun
Nori is like anything, the more you pay the better the quality.
The better the quality, means its usually a bit thinner and consumes better.
Cheap seaweed will be a bit thicker and once the moisture gets in it, will be chewy.
Generally, the Japanese seaweed is much better than chinese made seaweed, but as its about 4 times the price, most supermarkets only stock chinese.
They figure people wont know any better....
They may be right.
Also consume within a few minutes of making and the taste experience will be so much better.
cooking.stackexchange.com/users/34290/Bruce 
I use a propane blowtorch to crisp up my sushi.
If you try this, expect to ruin a few sushi rolls before you get it right.
cooking.stackexchange.com/users/51574/black toad
When we make home-made sushi, we often find that the seaweed wrapping (or nori) is too chewy.  The seaweed we buy is from the local supermarket, and nothing fancy.   Question : How can we ensure that the seaweed is not too chewy?
How does one stop sushi seaweed wrapping being too chewy?
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cooking.stackexchange.com/questions/19040
cooking.stackexchange.com/users/8046/Douglas S. Stones
Nori needs to be toasted to be crispy, and will be sold pre-toasted or will need to be toasted over an open flame. The quality of more expensive brands will be better, but in all cases the nori will lose its crispiness soon after coming in contact with the moist rice. Eat the sushi as the rice soaks into the nori very quickly and makes it chewy.
Nori should be toasted to  be crispy. Eat it quickly after making because the moisture from the rice soaks into the nori. Your rice may be too moist. Buy better quality nori. Try a propane blowtorch to crisp up the sushi.