Ok, let me rephrase your description and make sure I understand what you have. The 2.25 x 4.25 x 30' members are rafter ties to stabilize your roof from bowing in or out at the center of your rafter span. Attached to the bottom of these are some 3/4" (?) "straps" running perpendicular to your rafter ties to which your ceiling tiles are stapled. I am assuming probably roughly 12" x 12" tile size. Now you want to remove the tile, the strapping and every other rafter tie to open up the area and create a cathedral ceiling effect?
If I understand your intentions correctly, taking down the tiles and most of the strapping should not be a problem. I would not remove all the strapping because I would be concerned about a 30' long rafter tie wanting to bow sideways from the compression of roof load. A 2x4 or 2x6 could be added on top of the rafter ties to keep them from bowing. I would break the 30' span into at least thirds and run two rows the complete length of the roof similar to stabilizing the bottom chord of a truss.
I am like others here who are amazed that the rafter ties have not sagged. Are there any verticle supports going to the rafters anywhere along this 30' length? Also, what is the pitch of your roof and what is size and span of your roof rafters? These will make a difference in what you can do. If you decide to try to elimnate every other rafter tie, you will definitely need structural expertise. From an appearance standpoint instead of LVL's you might think about double 2 x ?'s (whatever the load requires) with spacer's which you can slide over both sides of the existing rafters and bolt through. In other words, two 2 x 8'2 (?) with a 2.25" spacers which you can fit over the rafters at the ends and bolt through. This approach might also eliminate the need for horizontal stabilization. Hope that is a little clearer than mud.
One more headache. Be careful when removing ceiling tile. I will be willing to bet that they contain asbestos.
The are the right age for it.