If it were my problem I would definitely start with isopropyl alcohol and use it to wet an absorbent cloth (thick cotton sheet or towelling). If you have difficulty finding isopropyl alcohol then have a look at eBay - one litre should be enough, it is an excellent, non-aggressive degreaser. Then once it has mostly dried follow with mild soap and water. Depending on how delicate the fabric is, use either a firm cloth (e.g. towelling) or a nail brush to wash away the remaining grease free dirt and then dab firmly with dry towelling to pick up the moisture together with residual dirt. Repeat if you think necessary.