Hi, 

I wonder if there is a way for the body to store fat without triglycerides accumulating in the blood.

Off the top of my head, I'd say lose some weight. Choose fewer complex carbohydrates from grains, and fewer fats to control caloric intake, and exercise a little more.

Eat less sugars. Lipid transfer through the bloodstream depends on lipoproteins (similar to the ones that cart your cholesterol around), and higher insulin levels are known to cause receptors to withdraw requests for lipids from the bloodstream, causing them to be stored or even fall to thicken arterial walls. Like cholesterol, we get triglycerides in two ways: from food we eat, and by manufacturing it ourselves cellularly.

Have your doctor check liver, kidney, and thyroid functions, and also any medications you may be taking (beta-blockers can increase triglycerides).