Viscosity is the resistance of a liquid to movement. Think water versus honey, honey being very viscous. Turns out a number of things can make the blood flowing through our bodies more or less viscous.
Some possible causes of increased blood viscosity are dehydration , psychological stress , cold temperatures , heat stroke , increased hematocrit (serum red blood cell) levels , hyperglycemia , increased triglycerides and chylomicrons in the blood , and perhaps exposure to air pollution  and tobacco smoke .
Based on that I would bet staying hydrated, avoiding stress, avoiding high hematocrit levels, preventing hyperglycemia and high triglycerides, and avoiding exposure to air pollution, smoking, and extreme temperatures would all be valid ways of avoiding thick blood.
Finally, There’s also a link between high blood pressure and high blood viscosity, likely relating to fibrinogen . Fibrinogen is increased by inflammation, so I suspect most methods of reducing inflammation would also improve blood viscosity. Omega-3 supplements, which have been shown to decrease inflammation, has been shown to improve blood viscosity .
Of course, I haven’t even said whether increased blood viscosity may increase risk of diseases, so let me get to that.
Several studies have found an association between heart disease and blood viscosity  leading some to suggest increased BV may be a causal factor in the development of the disease. The Edinburg Artery Study also noticed this correlation , but when controlling for a number of typical risk factors like smoking and BMI this association left the building. That’s the problem; because many things that are bad for your heart like smoking, inflammation, and high blood iron also increase blood viscosity, increased BV is likely to be correlated with heart disease. This is called a spurious relationship . Essentially, when C causes A and B it may appear that B is the cause of A since they will be seen together. Blood viscosity may increase heart disease risk, but I haven’t seen convincing evidence for it yet.
It’s also been noticed that people who’ve had strokes also [tend to have more viscous blood]. Elevated hematocrit and fibrinogen (the latter increased by inflammation) are factors increasing stroke risk and they both increase blood viscosity. A study on people with high levels of paraprotein , a condition that results in higher blood viscosity, found that higher BV might decrease cerebral blood flow, thereby increasing stroke risk (this study controlled for hematocrit). However, an [interesting experimental German study] concluded that blood viscosity itself does not appear to notably decrease cerebral blood flow. This is clearly an issue under debate, as this article  outlines and I don’t know the answer myself.
Ultimately I think high blood viscosity itself *might* increase risk of heart attack and stroke, but based on the evidence I think it’s more likely that the main causes of high blood viscosity also increase risk of heart attack and stroke and blood viscosity itself is just a marker for these risk factors.
High blood viscosity itself may decrease oxygen transport though  which could cause increased fatigue and a few other uncomfortable symptoms.