Always sand with the grain?

I am refinishing my oak coffee table which has quite a few inlays that are positioned with the grain at 90 degrees. What is the best method of sanding? It would be very tedious to sand every piece with the grain. Could I use a dual action sander with about a 360 grit to sand the entire table top?

A random orbital sander would now appear to be the only way to go.

A cabinet scraper is THE tool for the job - if you've got contrasting timbers it won't shove all the lighter dust into the darker bits.

I would think starting with 220 would be better, then go to 360, 400 and as far up as you like from there. As Bob said, the random obital , preferably with dust extraction would be my choice.

If you have contrasting timbers, do as Zitan says, go with the scraper. Sanding will make the light coloured timbers look very muddy. If the inlays are all the same timber, just at different angles, go with Bob & Gumby's recommendation.

Thanks for your help guys. I did start with 80 grit, then 120, then 180. I do believe my dual action sander is random orbital. All the inlays are the same oak as the rest on my table.