Is there a cd player that sounds close to an LP?


I am not sure what to do at this point. My system sounds great when I play my turntable.
I have a Marantz CD-67SE CD player, not high end by any means, but not a bad lower end player, so I thought. When I play CDs the brightness actually hurts my ears when I turn the volume up a bit.
I have been considering buy new cables for the speakers, maybe control this brightness, MIT T2's. Then was thinking, hate to ruin the sound I currently have from my platter. The highs and bass are perfect.
I have been considering buying an older tube CD player, in search of a warmer sound. I am thinking an Anthem CD1, a Conrad Johnson DV2B, maybe a Sonic Frontiers SFCD1. Not really sure what to buy.
Am I on the right track? I hate to change the sound of my system for CD's, and take away from the sound of my lp's.
Is there really a CD player out there that sounds even close to a LP? Maybe one that I might be able to pick up used for about a grand?
Any suggestions here, sure appreciate it!

Thank you.
johnymac
A couple of recommendations: a tube-equipped CD-player such as the Shanling TCD-200, or upgrading your existing player with a DAC such as Musical Fidelity's or Chord's DAC64. I went for the Chord, and like the sound very much. As urged above: Go and hear for yourself. I think the rigid affirmations "CDs never can sound as good as LP" are no longer true. There's by now a lot of excellently produced, or remastered CDs (not SACD, DVD-Audio) that, given goog CD-players, sound excellent, sometimes better than the old LPs. Good examples are a lot of remastered CDs from the EMI "Great Recordings of the Century"-series. Then, there is the XRCD-series from JVC. In Germany, even high-end-die-hard-vinyl-pope Attila Csampai was forced to admit that they sounded better than the original LPs on his state-of-the-art turntable system. I did a vinyl-CD-comparison with a vinyl-fan, and even fooled him when playing from CD.
I recently acquired a Mark Levinson 390S which I play through an Audio Research Ref 2 Mk II preamp, ARC VTM200 monoblocks, and Acoustat 2+2 full range electrostatic speakers. These speakers will show ANY digital harshness at all if it's present, and THERE SIMPLY IS NONE with this CD player. I can listen to CD's for hours, and it even lulls me to sleep sometimes. I also listen to a lot of vinyl, especially new records I buy off of ebay that have never been played. The best LP's have a magic and "rightness" that the CD's do not, but this is the first CD player I've ever heard that I consider a legitimate audiophile product.