Let's be clear what we mean by weight. Suppose you are holding an accelerometer then you are weightless if the accelerometer reads zero and your weight is non-zero if the accelerometer reads some non-zero value $a$. Your weight is then simply $ma$ where $m$ is your mass.
So for example the accelerometer in the smartphone in my pocket currently reads 1g so my weight is around 65g (my mass is around 65 kilos). If I were inside the international space station my phone accelerometer would read zero and I would be weightless (as any number of videos from the ISS will show).
Now we've established this, it should be obvious that acceleration in a straight line and the centripetal acceleration in circular motion both result in a non-zero weight.
I would guess you're really interested in whether this weight is the same whether the acceleration is due to gravity or whether it's due to a change in velocity. The answer is that yes it is the same, and indeed this is (one form of) the Einstein equivalence principle. More precisely the equivalence principle tells that acceleration and gravity are locally equivalent.
For more on this you may want to read the answers to Is it always possible to determine whether or not one is accelerating?, or search this site for more.