We will find the number of straights including straight flushes
and then subtract the number of straight flushes, in case you
don't want to include those:

We choose the denominations any of these 10 ways:
A,2,3,4,5
2,3,4,5,6
3,4,5,6,7
4,5,6,7,8
5,6,7,8,9
6,7,8,9,10
7,8,9,10,J
8,9,10,J,Q
9,10,J,Q,K
10,J,Q,K,A

For each of those ways we choose the suit for
the lowest card in 4 ways. That's 10*4 ways
For each of those ways we choose the suit for
the next to lowest card in 4 ways. That's 10*4*4
For each of those ways we choose the suit for
the middle card in 4 ways. That's 10*4*4*4
For each of those ways we choose the suit for
the next to highest card in 4 ways. That's 10*4*4*4*4
For each of those ways we choose the suit for
the highest card in 4 ways. That's 10*4*4*4*4*4
ways or 10*4^5 = 10240

We now find the number of straight flushes 
(including Royal flushes).

We choose the denominations any of these 10 ways:
A,2,3,4,5
2,3,4,5,6
3,4,5,6,7
4,5,6,7,8
5,6,7,8,9
6,7,8,9,10
7,8,9,10,J
8,9,10,J,Q
9,10,J,Q,K
10,J,Q,K,A
We can choose the 1 suit they must all have to be 
a straight flush in 4 ways.

That's 10*4 or 40 straight flushes.

So the number of straights is 10240-40 = 10200

The number of 5-card poker hands is
52 cards choose 5 = 52C5 = 2598960.

So the probability of being dealt a straight that
is not a straight flush is 

10200/2598960 = 5/1274 (dividing top and bottom by 2040).

That's about 0.003924646781789638932496075353218210361067503924646781789

If you want the probability including the straight
flushes, then it's

10240/2598960 = 128/32487 (dividing top and bottom by 80).

That's about 0.003940037553482931634192138393819066087973650998861082894…

[They are the same rounded to 4 decimal places. 0.0039.  That's 
because straight flushes are so rare].

Edwin