If you just multiply everything (don't forget to multiply the rows, too... if multiplying by 3, 1 stitch equals a 3x3 stitch square) it will start looking pixelated. That's not necessarily a bad thing, but it's something to be aware of.
What you might want to do is graph it out first. See what it looks like on paper. You might be able to just eyeball where the extra stitches should go to make it less pixelated. You might also be able to figure out roughly how big it will be in the end.
Other things that will make it bigger are using thicker yarn (or double-stranding) and a bigger hook.
If you make it and it doesn't turn out as big as you wanted, you might want to consider your options for a border on the blanket. It might be cute to make up smaller versions (in other words, don't change the pattern as written) and sew them around the edge of the big one. Something simpler might be stripes or any number of squares or stitch patterns out there.
ETA: I graphed out one leaf of the shamrock in Excel at 3x and this is what it looks like:
(resizing removed some of the lines... just ignore the lines. ^_^)
I think a lot of those sharp edges would be smoothed out just by adding one stitch in each corner, if that makes any sense...(second edit because I'm a dope)